We suggest a new framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory P ZF which resembles ZF , and is suitable for mechanization. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they deﬁne in an absolute way, independent of the extension of the “surrounding universe”. The language of P ZF is type-free, and it reﬂects real mathematical practice in making an extensive use of statically deﬁned abstract set terms. Another important feature of P ZF is that its underlying logic is ancestral logic (i.e. the extension of ﬁrst-order logic with a transitive closure operation)